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N.S., R.W. and B.-O.D. acknowledge support from the Swiss National Science Foundation (PP00P2-163967 and PP00P2-190080). M.N.G. acknowledges support from MIT's Kavli Institute as a Juan Carlos Torres Fellow and from the European Space Agency (ESA) as an ESA Research Fellow. A.A.B., B.S.S.and I.A.S. acknowledge the support of the Ministry of Science and Higher Education of the Russian Federation under the grant 075-15-2020-780 (N13.1902.21.0039). L.D. is an F.R.S.-FNRS Postdoctoral Researcher. B.V.R. thanks the Heising-Simons Foundation for support. This publication benefits from the support of the French Community of Belgium in the context of the FRIA Doctoral Grant awarded to M.T. and E.J. acknowledges DGAPA for his postdoctoral fellowship. Y.G.M.C. acknowledges support from UNAM-DGAPA PAPIIT BG-101321. D.D. acknowledges support from the TESS Guest Investigator Program grant 80NSSC19K1727 and NASA Exoplanet Research Program grant 18-2XRP18_2-0136. We acknowledge support from the Centre for Space and Habitability (CSH) of the University of Bern. Part of this work received support from the National Centre for Competence in Research PlanetS, supported by the Swiss National Science Foundation (SNSF). Funding for the TESS mission is provided by NASA's Science Mission Directorate. We acknowledge the use of public TESS data from pipelines at the TESS Science Office and at the TESS Science Processing Operations Center. This research has made use of the Exoplanet Follow-up Observation Program website, which is operated by the California Institute of Technology, under contract with the National Aeronautics and Space Administration under the Exoplanet Exploration Program. Resources supporting this work were provided by the NASA High-End Computing (HEC) Program through the NASA Advanced Supercomputing (NAS) Division at Ames Research Center for the production of the SPOC data products. This paper includes data collected by the TESS mission that are publicly available from the Mikulski Archive for Space Telescopes (MAST). This work is based upon observations carried out at the Observatorio Astronomico Nacional on the Sierra de San Pedro Martir (OAN-SPM), Baja California, Mexico. We warmly thank the entire technical staff of the Observatorio Astronomico Nacional at San Pedro Martir in Mexico for their unfailing support to SAINT-EX operations, namely: E. Cadena, T. Calvario, E. Colorado, F. Diaz, A. Franco, B. Garcia, C. Guerrero, G. Guisa, F. Guillen, A. Landa, L. Figueroa, B. Hernandez, J. Herrera, E. Lopez, E. Lugo, B. Martinez, G. Melgoza, F. Montalvo, J.M. Nunez, J.L. Ochoa, I. Plauchu, F. Quiroz, H. Riesgo, H. Serrano, T. Verdugo, I. Zavala. The research leading to these results has received funding from the European Research Council (ERC) under the FP/2007-2013 ERC grant agreement nffi 336480, and under the European Union's Horizon 2020 research and innovation programme (grants agreements nffi 679030 and 803193/BEBOP); from an Actions de Recherche Concertee (ARC) grant, financed by the Wallonia-Brussels Federation, from the Balzan Prize Foundation, from the BEL-SPO/BRAIN2.0 research program (PORTAL project), from the Science and Technology Facilities Council (STFC; grant nffi ST/S00193X/1), and from F.R.S-FNRS (Research Project ID T010920F). This work was also partially supported by a grant from the Simons Foundation (PI: Queloz, grant number 327127), as well as by the MERAC foundation (PI: Triaud). PI: Gillon is F.R.S.-FNRS Senior Research Associate. TRAPPIST is funded by the Belgian Fund for Scientific Research (Fond National de la Recherche Scientifique, FNRS) under the grant PDR T.0120.21, with the participation of the Swiss National Science Fundation (SNF). M.G. and E.J. are F.R.S.-FNRS Senior Research Associate. This work makes use of observations from the LCOGT network. Part of the LCOGT telescope time was granted by NOIRLab through the Mid-Scale Innovations Program (MSIP). M.S.I.P. is funded by NSF. Some of the observations in the paper made use of the High-Resolution Imaging instrument(s) `Alopeke (and/or Zorro). `Alopeke (and/or Zorro) was funded by the NASA Exoplanet Exploration Program and built at the NASA Ames Research Center by Steve B. Howell, Nic Scott, Elliott P. Horch, and Emmett Quigley. Data were reduced using a software pipeline originally written by Elliott Horch and Mark Everett. `Alopeke (and/or Zorro) was mounted on the Gemini North (and/or South) telescope of the international Gemini Observatory, a program of NSF's OIR Lab, which is managed by the Association of Universities for Research in Astronomy (AURA) under a cooperative agreement with the National Science Foundation, on behalf of the Gemini partnership: the National Science Foundation (United States), National Research Council (Canada), Agencia Nacional de Investigacion y Desarrollo (Chile), Ministerio de Ciencia, Tecnologia e Innovacion (Argentina), Ministerio da Ciencia, Tecnologia, Inovacoes e Comunicacoes (Brazil), and Korea Astronomy and Space Science Institute (Republic of Korea). This research has made use of the NASA Exoplanet Archive, which is operated by the California Institute of Technology, under contract with the National Aeronautics and Space Administration under the Exoplanet Exploration Program. This research made use of exoplanet (Foreman-Mackey et al. 2021a,b) and its dependencies (Agol et al. 2020; Kumar et al. 2019; Astropy Collaboration 2013, 2018; Kipping 2013; Luger et al. 2019; Salvatier et al. 2016; Theano Development Team 2016). Additional use of software packages AstroImageJ (Collins et al. 2017) and TAPIR (Jensen 2013). ; Context. Thanks to the relative ease of finding and characterizing small planets around M-dwarf stars, these objects have become cornerstones in the field of exoplanet studies. The current paucity of planets in long-period orbits around M dwarfs makes such objects particularly compelling as they provide clues about the formation and evolution of these systems. Aims. In this study we present the discovery of TOI-2257 b (TIC 198485881), a long-period (35 d) sub-Neptune orbiting an M3 star at 57.8 pc. Its transit depth is about 0.4%, large enough to be detected with medium-size, ground-based telescopes. The long transit duration suggests the planet is in a highly eccentric orbit (e similar to 0.5), which would make it the most eccentric planet known to be transiting an M-dwarf star. Methods. We combined TESS and ground-based data obtained with the 1.0-meter SAINT-EX, 0.60-meter TRAPPIST-North, and 1.2-meter FLWO telescopes to find a planetary size of 2.2 R-circle plus and an orbital period of 35.19 days. In addition, we make use of archival data, high-resolution imaging, and vetting packages to support our planetary interpretation. Results. With its long period and high eccentricity, TOI-2257 b falls into a novel slice of parameter space. Despite the planet's low equilibrium temperature (similar to 256 K), its host star's small size (R-* = 0.311 +/- 0.015) and relative infrared brightness (K-mag = 10.7) make it a suitable candidate for atmospheric exploration via transmission spectroscopy. ; Swiss National Science Foundation (SNSF) ; European Commission PP00P2-163967 PP00P2-190080 ; MIT's Kavli Institute ; European Space Agency European Commission ; Ministry of Science and Higher Education of the Russian Federation 075-15-2020-780 (N13.1902.21.0039) ; Heising-Simons Foundation ; French Community of Belgium ; DGAPA ; Programa de Apoyo a Proyectos de Investigacion e Innovacion Tecnologica (PAPIIT) Universidad Nacional Autonoma de Mexico BG-101321 ; TESS Guest Investigator Program 80NSSC19K1727 ; NASA Exoplanet Research Program 18-2XRP18_2-0136 ; Centre for Space and Habitability (CSH) of the University of Bern ; Swiss National Science Foundation (SNSF) ; European Research Council (ERC) 336480 ; Actions de Recherche Concertee (ARC) grant - Wallonia-Brussels Federation ; UK Research & Innovation (UKRI) ; Science & Technology Facilities Council (STFC) ; Science and Technology Development Fund (STDF) ST/S00193X/1 ; Fonds de la Recherche Scientifique - FNRS T010920F ; Simons Foundation 327127 ; MERAC foundation ; Fonds de la Recherche Scientifique - FNRS PDR T.0120.21 ; Swiss National Science Foundation (SNSF) ; National Science Foundation (NSF) ; NASA Exoplanet Exploration Program NASA's Science Mission Directorate ; European Research Council (ERC) 679030 803193/BEBOP ; Balzan Prize Foundation BEL-SPO/BRAIN2.0 research program (PORTAL project)